Animated Solution for Physics - Waves: The frequency of a car horn encountered a change from 400 Hz to 500 Hz, when the car approaches a vertical wall. If the speed of sound is 330 m/s, then the speed of car is …… km/h.

Imagine you are driving a car straight towards a massive vertical wall, and you press the horn. The sound travels forward, hits the wall, and bounces back to you as an echo. But here is the fascinating part: the pitch of the echo you hear is significantly higher than the pitch of the horn you just pressed! Why does this happen?

This is a classic manifestation of the Doppler Effect, but it happens not once, but twice in this scenario. Let's break down the physics of this double shift.

The Wall as an Observer

First, let's look at the sound traveling from the car to the wall. The car is a moving source emitting a frequency $f_0=400\text{ Hz}$, and the wall acts as a stationary observer. Because the source is moving towards the observer, the sound waves get compressed in front of the car.

The frequency received by the wall, let's call it $f^{\prime }$, is given by the standard Doppler formula:

Here, $v_s$ is the speed of sound ($330\text{ m/s}$) and $v$ is the speed of the car.

The Wall as a Source

Now, the wall reflects this exact frequency $f^{\prime }$ back towards the car. The wall is now a stationary source emitting frequency $f^{\prime }$. But the car isn't stationary; it is moving towards the wall!

The driver is now a moving observer heading towards a stationary source. This causes a second Doppler shift, increasing the frequency even further. The final frequency heard by the driver, $f^{\prime \prime }=500\text{ Hz}$, is:

If we substitute the expression for $f^{\prime }$ from the first phase into the second equation, something beautiful happens. The speed of sound $v_s$ in the numerator and denominator cancels out perfectly:

This combined formula is a highly potent tool for JEE physics. Whenever an observer moves towards a wall and hears an echo, you can use this directly!

Let's plug in our known values to find the car's speed $v$:

Dividing both sides by 100 gives:

Cross-multiplying to solve for $v$:

Finally, the question specifically asks for the speed in km/h. We must multiply by $\frac{18}{5}$:

The car is traveling at a brisk $132\text{ km/h}$!